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In 1979 Lusztig proposed a conjectural construction of supercuspidalrepresentations of reductive p-adic groups, which is similar to the well knownconstruction of Deligne and Lusztig in the setting of finite reductive groups.We present a general method for explicitly calculating the representationsarising from Lusztig's construction and illustrate it with several examples.The techniques we develop also provide background for the author's joint workwith Weinstein on a purely local and explicit proof of the local Langlandscorrespondence.

Deligne-Lusztig constructions for unipotent and p-adic groups